|
Search: id:A005369
|
|
|
| A005369 |
|
a(n) = 1 if n is of form m(m+1) else 0. |
|
+0
4
|
|
| 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
(list; graph; listen)
|
|
|
OFFSET
|
0,1
|
|
|
COMMENT
|
Euler transform of period 4 sequence [0,1,0,-1,...].
Expansion of q^(-1/4)eta(q^4)^2/eta(q^2) in powers of q.
|
|
FORMULA
|
G.f.: Product_{k>0} (1-x^(4k))/(1-x^(4k-2)) = f(x^2, x^6) where f(a, b) is Ramanujan's theta function.
Given g.f. A(x), then B(x)=(x*A(x^4))^2 satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=v^3+4*v*w^2-u^2*w. - Michael Somos Apr 13 2005
Given g.f. A(x), then B(x)=x*A(x^4) satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)=u1*u2^2*u6-u1*u6^3-u3^3*u2. - Michael Somos Apr 13 2005
a(n)=b(4n+1) where b(n) is multiplicative and b(2^e)=0^e, b(p^e)=(1+(-1)^e)/2 if p>2. - Michael Somos Jun 06 2005
|
|
EXAMPLE
|
1 + q^2 + q^6 + q^12 + q^20 + q^30 + q^42 + q^56 + ...
|
|
PROGRAM
|
(PARI) a(n)=if(n<0, 0, issquare(4*n+1))
(PARI) a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff(eta(x^4+A)^2/eta(x^2+A), n))
|
|
CROSSREFS
|
Cf. A002378. Partial sums give A000194. A010054(n)=a(2n).
Adjacent sequences: A005366 A005367 A005368 this_sequence A005370 A005371 A005372
Sequence in context: A016334 A087032 A102242 this_sequence A108340 A088917 A014933
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
njas
|
|
EXTENSIONS
|
Additional comments from Michael Somos, Apr 29 2003.
|
|
|
Search completed in 0.002 seconds
|
